Smulaton and optmzaton of supply chans: alternatve or complementary approaches? Chrstan Almeder Margaretha Preusser Rchard F. Hartl Orgnally publshed n: OR Spectrum (2009) 31:95 119 DOI 10.1007/s00291-007-0118-z Abstract Dscrete-event smulaton and (mxed-nteger) lnear programmng are wdely used for supply chan plannng. We present a general framework to support the operatonal decsons for supply chan networks usng a combnaton of an optmzaton model and dscrete-event smulaton. The smulaton model ncludes nonlnear and stochastc elements, whereas the optmzaton model represents a smplfed verson. Based on ntal smulaton runs cost parameters, producton, and transportaton tmes are estmated for the optmzaton model. The soluton of the optmzaton model s translated nto decson rules for the dscrete-event smulaton. Ths procedure s appled teratvely untl the dfference between subsequent solutons s small enough. Ths method s appled successfully to several test examples and s shown to delver compettve results much faster compared to conventonal mxed-nteger models n a stochastc envronment. It provdes the possblty to model and solve more realstc problems (ncorporatng dynamsm and uncertanty) n an acceptable way. The lmtatons of ths approach are gven as well. Keywords Supply chan management Optmzaton Dscrete-event smulaton Hybrd method C. Almeder (B) M. Preusser R. F. Hartl Unversty of Venna, Brünnerstr 72, 1210 Venna, Austra e-mal: chrstan.almeder@unve.ac.at M. Preusser e-mal: margaretha.preusser@unve.ac.at R. F. Hartl e-mal: rchard.hartl@unve.ac.at H.O. Günther, H. Meyr, Supply Chan Plannng c Sprnger-Verlag Berln Hedelberg 2009 29
30 C. Almeder et al. 1 Introducton In recent years ntra-company supply chans have been growng sgnfcantly spannng producton and dstrbuton stes all over the world. At the same tme global competton has ncreased, such that there s a strong demand for new decson support tools on strategc, tactcal and operatonal levels. Bswas and Narahar (2004) classfed the relevant research on such decson support systems nto three categores: (a) (b) (c) Optmzaton models manly for mult-echelon nventory control. In most cases these models are determnstc and used for strategc or tactcal decsons. Analytcal performance models, whch consder a dynamc and stochastc envronment. They are used to nvestgate desgn or prncpal management decsons. Such systems are represented as Markov chans, Petr nets or queung models. Smulaton and nformaton models, whch are used to analyze complex dynamc and stochastc stuatons and to understand ssues of supply chan decson makng. For the frst and the second categores, t s often necessary to make several smplfcatons from the real-world case n order to develop solvable models. Nevertheless the problem sze s usually very lmted. Although there are promsng developments of combnatons of these two categores (cf. Sect. 2), many of them reman on a strategc level and the stochastc property s consdered by a small number of dfferent scenaros. In ths paper we develop a new soluton approach by applyng a LP/MIP formulaton n the context of a dscrete-event smulaton. So we are able to combne the advantages of models from all categores mentoned above by consderng a detaled representaton of a dynamc and stochastc envronment and allow the applcaton of optmzaton methods n ths context. Our nvestgatons are based on a general supply chan network model wth dfferent facltes (supplers, manufacturers, dstrbutors) and dfferent transportaton modes connectng these facltes. We assume that there s a central planner wth perfect nformaton such as for ntra-company supply chans or supply chans wth a domnant member. Ths problem settng s motvated by a case study about a global supply chan network n the paper ndustry (Gronalt et al. 2007). The goal s to reduce costs by smultaneously optmzng the producton/transportaton schedule and reducng nventory levels. We are amng for a robust soluton, n the sense that a stochastc envronment s consdered. Comparng our problem to the tasks n the supply chan matrx (cf. Stadtler 2005), the problem s a combnaton of several operatonal tasks: producton plannng, dstrbuton plannng and transport plannng. In addton to other approaches, we assume a stochastc smulaton model for these tasks and combne t wth classcal optmzaton approaches. Our goal s to acheve an optmal operaton plan for supply chan networks by combnng optmzaton models and smulaton models. We do not use the optmzaton on top of the smulaton, where an optmzaton algorthm uses the smulaton model as a black-box evaluaton functon (cf. Glover et al. 1999). Instead we nclude smulaton and optmzaton n an teratve process n order to gan the advantages of optmzaton (exact soluton) and smulaton (nonlneartes, complex structure, stochastcty). In the prevous research (Almeder and Preusser 2004; Preusser et al. 2005a,b) we
Smulaton and optmzaton of supply chans 31 smulate Solutons of smulaton experments aggregate Decson rules n D-E model Optmzaton model derve Lnear soluton optmze Fg. 1 Interacton between smulaton and optmzaton developed a rough dea of ths concept. In the current paper we extended ths concept, such that t s possble to apply t to a wde range of supply chan problems. Furthermore we analyze n detal the advantages and dsadvantages of ths approach and present results for dfferent test cases. The supply chan s represented as a dscrete-event model (D-E model) and a smplfed verson s modeled as an optmzaton model. We start by performng several smulaton runs n order to get average values of the parameters (e.g., unt transportaton costs) whch are then fed nto the optmzaton model. After solvng the optmzaton model the result s transformed nto decson rules that are used n the dscrete-event model. Then we start agan wth further smulaton experments (see Fg. 1), and so on. Our contrbuton s twofold: Development and analyss of a general framework (Fg. 1) and a toolbox for the combnaton of dscrete-event smulaton and optmzaton of supply chans. For stochastc supply chans, an teratve combnaton of smulaton and lnear programmng s emprcally shown to be compettve compared to determnstc MIP-models. The paper s organzed as follows: We start wth a lterature revew n Sect. 2, followed by a descrpton of the general model framework n Sect. 3. In Sect. 4 we explan the lnkage between the smulaton model and ts lnear verson. Fnally, we report on dfferent test results n Sect. 5 and gve conclusons and an outlook for possble future research n Sect. 6. 2 Lterature revew 2.1 Supply chans Aspects of the ntegraton of transport and producton plannng wthn supply chans have been nvestgated n several papers (cf. Erengüc et al. 1999). Combned plannng approaches for dfferent decson levels (e.g., tactcal and operatonal decsons) can be found n Meyr (2002) and Schneewess (2003). There are numerous papers dealng wth lnear or mxed-nteger programs for supply chan networks and network flows. Yaged (1971) dscussed n hs paper a statc network model whch ncludes
32 C. Almeder et al. nonlneartes. He tres to optmze the product flow by solvng a lnearzed verson of the network and to mprove the flow n the network. Paraschs (1989) dscussed several dfferent possbltes to lnearze such networks and Fleschmann (1993) presented several applcatons of network flow models, whch are solved through lnearzaton. Pankaj and Fsher (1994) showed that based on an MIP model the coordnaton of producton and dstrbuton can reduce the operatng cost substantally. Dogan and Goetschalckx (1999) showed that larger supply chan desgn problems can be solved usng decomposton. A recent case study about a supply chan of the pulp ndustry modeled as a MIP s gven by Gunnarsson et al. (2007). In general the problems solved wth LPs and MIPs usually nclude several smplfcatons n order to keep them solvable. Recent publcatons also ncluded stochastc elements n the optmzaton models. Santoso et al. (2005) consdered a stochastc programmng approach for the supply chan network desgn. They used a sample average approxmaton and Benders decomposton to solve desgn problems for a supply chan whle consderng future operatonal costs. For that purpose they developed a lnear model wth uncertan cost factors and demand. Although they used a fast algorthm, realstc problems wth sample szes of up to 60 scenaros need several hours to be solved. Alonso-Ayuso et al. (2003) consdered a smlar combned desgn and operaton problem. Ther stochastc programmng approach was able to solve medum szed problems wth about 100 bnary decsons wthn 1 h. Leung et al. (2007) presented a robust optmzaton model for a smultaneous producton plannng for several stes n a supply chan under uncertanty. But stll they are restrcted to rather small models and consder only four dfferent scenaros. In the feld of supply chan smulaton Klejnen (2005) gave a short overvew of smulaton tools and technques used for supply chans. He dstngushed between four dfferent approaches: spreadsheet smulaton, system dynamcs, dscrete-event dynamc systems smulaton, and busness games. Clearly, dscrete-event smulaton s the most powerful tool to consder complex stochastc systems. Numerous software packages for dscrete-event smulaton are avalable, both very specalzed ones for a specfc part of the supply chan and general ones wth a hgh functonalty n modelng and vsualzaton of supply chans (cf. Kelton et al. 2002; Kuhn and Rabe 1998). One example s the Supply Net Smulator presented by Stäblen et al. (2007). It allows smulatng the behavor of ndvdual members n a supply chan network. They used an agent-based approach, where each member optmzes ts own operatons n the sense of an advanced plannng system. But there s no nteracton between smulaton and optmzaton. 2.2 Optmzaton and smulaton Most of today s smulators nclude possbltes to do a black-box parameter optmzaton of a smulaton model. Glover et al. (1999) presented the successful development of OptQuest ( OptTek Systems, Inc. 1 ), an optmzaton toolbox contanng dfferent 1 http://www.opttek.com.
Smulaton and optmzaton of supply chans 33 algorthms (manly metaheurstcs) desgned to optmze confguraton decsons n smulaton models. The smulaton model s used only for the evaluaton of the objectve value, no further structural nformaton s used. Swsher et al. (2000) and Fu (2002) stated n ther papers that there s stll a bg gap between optmzaton methods for smulaton-based optmzaton used n commercal software and methods avalable n research lterature. Truong and Azadvar (2003) developed an envronment for solvng supply chan desgn problems, where they combne smulaton wth genetc algorthms and mxednteger programs. Strategc decsons regardng faclty locaton and partner selecton are consdered. The work by Lee and Km (2002), possbly the most related work n ths context, shows a combnaton of smulaton and optmzaton for the case of a productondstrbuton system. They use smulaton to check the capacty assumptons used for a smpler lnear model n a more realstc envronment wth stochastc machne breakdowns and to update these capacty parameters for the optmzaton. After several teratons they end up wth a soluton of the optmzaton model whch s also wthn the constrants of the stochastc smulaton model. Ther method s qute smlar to our approach, but they am for more realstc capacty estmaton for the optmzaton model. In contrast, we try to fnd a robust plan for producton, stockng, and transportaton consderng stochastc and nonlnear operatons and costs by estmatng delays and cost factors based on smulaton experments. 3 The supply chan network model The general descrpton of the supply chan orgnates from a case study about a supply chan n the paper ndustry (cf. Gronalt et al. 2007). Several producton stes are used to manufacture dfferent paper products, whch are delvered ether drectly or va hubs to customers all over the world. The man task n ths case study was to develop a 1-year plan for producton quanttes and transportaton lnks. In ths case study a statc model was developed, whch was used to get rough estmates quckly. Inspred by ths case study we formulate the followng problem settng. The bass for our supply chan model s a predefned network,.e., the locatons of all actors and the connectons between them are gven. Wthn the network we dfferentate between three types of partcpants connected by transportaton lnks: supplers provdng raw materals; customers who demand certan products at a specfc tme; producton/warehouse stes where producton, stockng, and transshpment takes place. The whole supply chan s order-drven, that means products are manufactured or transported only f a subsequent member of the supply chan requests t. So the orgn for all actvtes s the predefned determnstc demand of the customers. All actvtes are based on tme perods, whch mght be days or shorter tme perods. The supplers are used as source for raw materals, whch are sent to producton stes f requested. Producton/warehouse stes can store ncomng products. These products
34 C. Almeder et al. can be used to manufacture new products, or they are smply transferred to the output nventory. From there they are sent to subsequent members of the supply chan. The smulaton model s mplemented usng AnyLogc ( XJ Technologes), a Java-based smulaton tool. The model s constructed as a lbrary ncludng several dfferent modules. These modules represent the four types of partcpants n the supply chan network plus a general control module necessary for controllng the smulaton experments as well as the communcaton wth the optmzaton model whch was developed usng Xpress-MP ( Dash Optmzaton). Ths model s a smplfed determnstc verson of the lbrary modules of the smulaton model. In ths secton we wll explan the dfferent modules of our supply chan network. 3.1 Module suppler Smulaton model. Ths module s used to generate certan products, store them, and delver them f demanded. It has one nput port to receve orders for products and one output port to delver products. If ths module receves an order through the nput port, then t sends the requested amount of products va the output port. If the amount exceeds the current nventory level, only the avalable amount s sent. As soon as new products arrve n the nventory they are delvered untl the whole order has been fulflled. The costs arsng n ths module are only nventory costs for storng products pror to delvery. These costs may have any user-defned functonal form. Accordng to the gven parameters n each perod, new products are generated and added to the stock. Optmzaton model. We also developed a smplfed representaton as an optmzaton model. We denote by J S the set of supplers wthn a network, by P the set of products, and by T the number of perods. The representaton of the suppler s behavor n the optmzaton model can be formulated as follows (If p and t are free ndces,.e. not used as a summaton ndex, then the set of equatons s meant to be vald t = 1,...,T, p P.): TC S = T out H p ( out l p (t)) J S, (1) p P t=1 (t) = out l p (t 1) out f p (t) + S p (t) J S, (2) out l p (t) 0 J S. (3) out l p For a complete lst of parameters and varables, see Appendx C. The overall cost of suppler s denoted by TC S, consstng only of the holdng cost out H p ( ) of the output nventory, expressed by the rght-hand sde of (1). Equatons (2) are the nventory balance equatons for the output nventores out l p (t). The stock s dmnshed by the outflow of materals, out f p (t), and ncreased by the gven supply S p (t). Thelast set of constrants (3) guarantees that the nventory level cannot be negatve. The smulaton and the optmzaton model are connected va the holdng costs n (1) whch represent the user-defned cost functon n the smulaton.
Smulaton and optmzaton of supply chans 35 3.2 Module producton Smulaton model. Ths module s the core of the whole model. It represents a producton ste as well as a transshpment pont. It conssts of an nput and an output storage. Items are ether transformed nto new tems or smply transferred from the nput to the output storage. Ths module has one nput port and one output port for orders, as well as one nput and one output port for products. The nput storage s replenshed by orderng products va the output port for orders from a suppler or another producton module. The orderng polcy may be ether autonomous (e.g., an (s,s)-polcy or any user-defned polcy) or t s determned by the result of the optmzaton model. Products are receved through the product nput port and stored n the nput nventory. The producton of new products or the transfer of products s ntated by an nternal order placed by the output nventory (ether autonomous or based on the soluton of the optmzaton model). The delay for producton and transfer s a user-defned functon. It may contan stochastc elements and depend on other parameters (e.g., the current load). Producton and transfer have lmted capactes and furthermore producton s restrcted to the avalablty of raw materals (other products). If these capactes do not allow producng (or transferrng) a lot as a whole, t s splt nto several batches. Through the nput order port the module receves orders from other producton or customer modules. Products are sent through the output product port accordng to these orders and based on avalablty. Costs arse n ths model for nventory holdng (nput and output), for producton, and for transfer. Optmzaton model. The optmzaton model for the producton node s as follows (we denote by J I the set of producton nodes n the supply chan network): TC I = p P + p P T t=1 W p ( p m (t)) + p P T t=1 m p (t) prod Cap p (t), u p (t) ta Cap p (t), T t=1 ( ) n H p n l p (t) + p P p P p P n l p (t) = n l p (t 1) + n f p (t) Z p ( p u (t)) T t=1 out H p ( out l p (t)) J I, (4) a p m p (t) prod C (t) J I, (5) d p u p (t) ta C (t) J I, (6) α p (p ) m p (t) u p (t) + r p (t) J I, (7) p P out l p (t) = out l p (t 1) out f p (t) + χ p t δ m p (t δ p ) +χ p t σ u p (t σ p ) + s p (t) J I, (8) n l p (t) nvn Cap p (t), q p n l p (t) n L (t) J I, (9) p P
36 out l p (t) nvout Cap p (t), p P q p m p (t) 0, u p (t) 0, n l p (t) 0, out C. Almeder et al. out l p (t) out L (t) J I, (10) l p (t) 0 J I. (11) The overall cost of a producton node s represented by TC I. These costs consst of producton costs (the producton amounts are denoted by m p (t)) transfer costs (the transfer amounts are denoted by u p (t)) and the holdng costs of the nput and the output nventory. Constrants (5) and (6) restrct the producton and the transfer for each product ndvdually, as well as for the total producton and transfer. In the latter case the amounts are multpled by the resource requrements. The dstncton between ndvdual capacty constrants for each product and global capacty constrants are necessary to cover general stuatons where dfferent resources as well as common resources are necessary for producton. Equatons (7) are the nventory balance equatons for the nput nventores. The current nventory level s determned by the nventory level of the prevous perod, the nflow from other nodes, the requred raw materals for producton, the transfer amount and some external nflow (from outsde of the system); α p (p ) represents the unts of raw materal p whch s necessary to produce one unt of product p. The nventory balance equatons for the output nventores (8) are smlar, but the producton and transfer delays (δ p,σp )havetobe consdered before a new product arrves n the output nventory. Functon χ t ε s an ndcator functon, used n order to avod the use of producton and transfer amounts for negatve perods. Equatons (9) and (10) are used to restrct the stock of the nput and the output nventory (for each product separately and accumulated usng the space requrements q p ). These two types of restrctons allow modelng a dedcated-storage as well as a random-storage polcy. The smulaton model and the optmzaton model are connected through the cost factors n Eqs. (4) and producton and transfer delays (δ p,σp ), whch are user-defned functons n the smulaton model possbly contanng stochastc and nonlnear elements. Furthermore the producton and transfer amounts (m p (t), u p (t)) ofthe optmzaton model are used to determne producton plans n the smulaton model. For example, f m p (t) >0 for a specfc product and perod, then n the smulaton model the amount gven by m p (t) s ordered n perod t. 3.3 Module customer Smulaton model. Accordng to a gven demand table, the customer places orders at the producton stes. Due to stochastc features wthn the smulaton, t s not possble to tme delveres exactly. Therefore the customer has an nput nventory, whch s used to satsfy the demand. The nventory level can be negatve (shortages), as well as postve (oversupply). In both cases penalty costs occur, whch are hgher for shortages. The module has one output port for sendng requests and one nput port for recevng products. The orders are sent ether accordng to the demand table (ncludng a standard delay tme for transportaton) or accordng to the soluton of the optmzaton model.
Smulaton and optmzaton of supply chans 37 Optmzaton model. The optmzaton model for the customers behavor can be wrtten as follows (we denote by J C the set of customer nodes n the supply chan network): TC C = p P T t=1 R p (n b p (t)) J C, (12) n l p (t) n b p (t) = n l p (t 1) n b p (t 1) + n f p (t) D p (t) + r p (t) J C, (13) n l p (t) = 0 J C. (14) In (12) we calculate the cost at the suppler whch conssts only of penalty cost for back orders n b p (t). Equatons (13) are the nventory balance equatons where the customers demands D p (t) are consdered. It s assumed that all customers are justn-tme customers. Therefore, constrants (14) ensure that no oversupply (postve stock level) s possble,.e. t s not allowed to send more products than demanded by the customers. Ths JIT assumpton may be dropped and holdng costs for postve stock may be ncluded. In the smulaton model the JIT assumpton s weakened, because stochastc transportaton tmes may cause an unwanted early delvery. These early delveres are penalzed. The dfferences between the smulaton model and ts representaton as an optmzaton model are the penalty cost factors n (12) and the JIT assumpton expressed n (14). 3.4 Module transport Smulaton model. Ths module s used to transport products between dfferent modules. It receves products through ts sngle nput port and sends t (accordng to some tme delay) through the output port to the next module (Producton or Customer). It has a lmted capacty and organzes the transports accordng to a FIFO rule. It s also possble to splt shpments f the avalable capacty does not allow sngle shpment. The tme delay may be stochastc and may depend on other parameters. User-defned costs arse for transportaton and may nclude transportaton tme, amounts, and fxed charge parts. Optmzaton model. The representaton of the transport modules as an optmzaton model can be formulated as follows. Each transport module s dentfed by the ndces of the nodes, whch t connects. Furthermore, we need an addtonal ndex v V denotng the dfferent transport modes (f v s not used as a summaton ndex, the equatons are vald for all v V ): TC T j = T t=1 v V p P ( ) v C p v j x p j (t) J S J I, j J I J C, (15)
38 C. Almeder et al. v x p j (t) v Cap p j (t), v g p vx p j (t) v C j (t) J S J I, j J I J C, p P (16) n f p j (t) = out f p (t) = J s J I v τ j <t v V j J I J C v V v x p j (t v τ j ) j J I J C, (17) v x p j (t) J S J I, (18) v x p j (t) 0 J S J I, j J I J C. (19) The total transportaton cost for transports from member to member j of the supply chan network s denoted by TCj T and v x p j (t) gves the transportaton amount for each perod, each product, and each transportaton mode. Constrants (16) lmt the transportaton to a product-specfc and an overall capacty lmt. Equatons (17) and (18) represent the nflow of products to member j and the outflow of products from member. The connecton between the smulaton and the optmzaton model s establshed by the transportaton cost functons n (15) and the transportaton delays v τ j n (17) on the one hand, and through the transportaton amounts v x p j (t) on the other hand. These transportaton amounts are used to defne orderng schemes for the Producton and Customer modules n the smulaton model. That means f v x p j (t) >0, then member j of the supply chan sends a request for v x p j (t) unts of product p to member of the supply chan at tme t. 3.5 Supply chan optmzaton model The optmzaton model of the whole supply chan network s defned by mnmzng the total cost mn TC S + TC I + TC C + J S J I J C J S J I j J I J C TC T j (20) subject to the constrants (1) (19). If we assume that all cost functons are lnear,.e. that the objectve (20) s a lnear functon, we can wrte t as follows: mn j J p P + J I + t=1,..t v V p P t=1,..t J S J I p P t=1,..t v c p j vx p j (t) + J I z p u p (t) + J I p P t=1,..t p P t=1,..t out h p out l p (t) + w p n h p n l p (t) J C p P t=1,..t m p (t) ρ p n b p (t). (21)
Smulaton and optmzaton of supply chans 39 suppler producton customer transport_1 transport_2 param (LP) out p h param (LP) v v c, t p j DV (Sm) v x p (t) j j param (LP) p w,, p n h p d m p,, p z out p h s p DV (Sm) p (t), u (t), param (LP) v v c, t p j j DV (Sm) v p xj (t) param (LP) n p r Fg. 2 An example of the module confguraton for a smple supply chan consstng of one suppler, one producton ste, and one customer (dashed lnes ndcate nformaton flow and sold lnes ndcate materal flow). For the case of a lnear program the nformaton below the modules represents the parameters whch are calculated durng the smulaton runs and transferred to the LP, param (LP), and the decson varables of the LP used as decson rules n the smulaton model, DV (Sm.) Hence, we get a lnear programmng model whch we wll use n connecton wth the smulaton model as depcted n Fgure 1. A detaled descrpton of the lnear model can be found n Preusser et al. (2005a,b). Ths model s a pure lnear program whch can be solved easly wth any standard LP-solver wthn very short tme. If necessary, t s possble to extend the model formulaton to consder more features, e.g., fxed-cost transportaton, bnary decsons, step functons, etc. These extensons lead to a mxed-nteger formulaton, thus ncreasng computatonal tme (cf. Appendx A). 3.6 Supply chan smulaton model Implementng a smulaton model n AnyLogc means to arrange the accordng modules and connect them. In Fg. 2 an llustratve example shows, how these modules can be connected n order to mantan nformaton flow (drect connectons between Suppler, Producton, and Customer) and materal flow (va Transport modules). Furthermore the cost and delay functons for each module must be specfed. 4 Connectng the optmzaton wth the smulaton In order to couple the optmzaton model and the smulaton model, we frst have to defne the requred data and the way they should be exchanged. We decded to use an MS Access database to store all necessary nformaton whch s: General network structure: Ths ncludes the number of actors n the supply chan and the accordng lnks between them. General parameters used n the smulaton and optmzaton models: These sets of parameters nclude all capacty lmtatons, resource parameters, bll-of-materals, predefned supply at the supplers, and predefned demand at the customers.
40 C. Almeder et al. aggregated results (transportaton delays, producton delays, ) smulaton model general parameters MS Access database general parameters optmzaton model (AnyLogc) (ODBC) (Xpress) decson rules (orderng plans, producton schedules, ) Fg. 3 Ths scheme shows the data exchange between the smulaton and the optmzaton model va the MS Access database n the mddle Results of the optmzaton model (= parameters for the smulaton): The results of the optmzaton used as decson rules n the smulaton are producton and transfer quanttes, as well as transportaton amounts (m p (t), u p (t), v x p j (t)). Results of the smulaton model (= parameters for the optmzaton model): The man results of the smulaton experments used n the optmzaton model are the cost parameters and the delays for producton, transfer and transport. The smulaton model s desgned as the master process, whch controls the data communcaton and the LP/MIP-solver. The smulaton model and the optmzaton model retreve and store values from and to the database usng the Open Database Connectvty (ODBC) nterface (see Fg. 3). To ntate the optmzaton process n our system, a few smulaton runs are performed usng the data from the database. Mssng decson rules whch, n later teratons, are generated usng the results of the optmzaton model, are substtuted by autonomous decson rules (lke an (s,s)-polcy for the replenshment). These frst smulaton runs are only necessary to generate ntal parameter values for the optmzaton model, but ther results wll be gnored n further teratons n order to avod basng effects caused by the autonomous decson rules. The results of the ntal runs (delays, per unt costs, etc.) are aggregated and accordng means and varances are stored n the database (see Sect. 4.1). Afterwards Xpress-MP s executed. It loads the general data and the smulaton results from the database, computes the soluton of the optmzaton model and stores the results (orderng and delvery plans, producton and transfer schedules, etc.) n the database. Then we start agan wth fve smulaton experments usng now the newly computed decson rules (see Sect. 4.2), based on the soluton of the optmzaton model. Further on we wll denote ths algorthm by SmOpt (or SmLP, f a pure LP model s used for the optmzaton part, and SmMIP, f a mxed-nteger formulaton s used). In Table 1 a pseudo code of ths SmOpt algorthm s gven. 4.1 Aggregatng smulaton results Snce the smulaton model may contan stochastc and nonlnear elements, t s necessary to perform several smulaton runs and combne the results. For the cost parameters, necessary for a lnear optmzaton model, we calculate average per unt cost. That means, e.g., for the producton costs we accumulate the total cost for the
Smulaton and optmzaton of supply chans 41 Table 1 Pseudo code for the combned smulaton optmzaton approach SmOpt SmOpt: Load necessary smulaton parameters from the database Perform a few smulaton runs usng autonomous decson rules Aggregate results and store them n the database whle stoppng crtera are not met Load aggregated parameters nto LP/MIP-Solver Solve the optmzaton model Wrte new decson rules to the database Load new decson rules nto smulaton model Perform smulaton runs usng these decson rules Aggregate results and store them n the database end-whle whole plannng horzon for a certan product and dvde these costs by the number of products produced. Other parameters, called crtcal parameters, have a drect nfluence on the materal flow (e.g., transportaton delay). The use of average values for those parameters would most probably lead to bad results. In about half of the cases the delay would be longer than assumed and would cause addtonal delays n subsequent operatons. Therefore, t seems reasonable to use, e.g., a 90%-quantle (based on a normal dstrbuton wth estmated mean and varance calculated from the dfferent smulaton runs) for such delay parameters. Ths results n an overestmaton of the delays for the optmzaton model, because the tme s determned such that 90% of the observed delays wll be shorter, but t ensures that a smooth materal flow through the network s possble. For the crtcal parameters t s useful to combne results from the prevous teratons wth current ones, n order to enlarge the sample sze and to get better estmates of the mean and the varance. 4.2 Decson rules based on the soluton of the optmzaton model There are several possble ways to use the soluton of the LP-model wthn the smulaton model. One method, whch we apply here, s to use the transportaton, producton and transfer results ( v x p j (t), m p (t), u p (t)) as a gven orderng plan. Accordng to the transportaton values v x p j (t) module j sends a request to module at tme t for the gven amount v x p j (t) of products of type p usng the transportaton mode v. Smlarly, producton and transfer results can be used. In some cases, due to the stochastc features of the smulaton model, t may happen that some of the modules are outof-stock for a specfc product. Snce unfulflled orders are backlogged, these requests are fulflled as soon as the products are avalable. More complex procedures would be, e.g., to use results of the senstvty analyss (dual varables, reduced costs) to determne the crtcal parameters, to observe these parameters durng the smulaton runs, and to adapt decson rules (use a dfferent
42 C. Almeder et al. 310000 260000 smulaton optmzaton total cost 210000 160000 110000 60000 10000 1-Sm 1-Opt 2-Sm 2-Opt 3-Sm 3-Opt 4-Sm 4-Opt teraton Fg. 4 Objectve values of the optmzaton model and the smulaton model for each teraton for a determnstc model consderng fxed costs for producton, transfer and transport 5-Sm 5-Opt 6-Sm 6-Opt soluton of the optmzaton model) f the observed parameters reach a certan threshold. For our examples we use the frst approach for translatng the soluton of the optmzaton model nto decson rules for the smulaton model (cf. Sect. 3). In ths paper we wsh to nvestgate the drect nteractons between the soluton of the optmzaton model and the smulaton results. The analyss of more complex decson rules goes beyond the scope of ths paper and mght by a subject for further research. 5 Tests and results We wsh to nvestgate the followng research questons wth emprcal tests usng a set of test nstances: Does ths method converge n practce for realstc test cases? If we can observe convergence, s the result optmal or at least a good approxmaton? Is ths method advantageous compared wth tradtonal plannng methods? Although t s not possble to prove general convergence for all our test nstances, we observe fast convergence of the objectve values of the smulaton and optmzaton model. Fgure 4 shows a typcal stuaton usng the results of the determnstc test nstance D1-L descrbed n Sect. 5.1. We start wth the smulaton model usng an autonomous rule for replenshng the nventores. Snce we start wth all nventores empty, t takes a long tme, untl the orders are fulflled. Ths causes hgh costs and an overestmaton of transportaton and producton delays. Therefore, the frst soluton of the lnear model leads also to a hgh objectve value manly consstng of penalty costs for late (or even no) delveres. Consequently, the smulaton model leads to a smlar objectve functon n teraton 2, because t uses the delvery plans of the soluton of the lnear model. Due to the fact that the soluton of the lnear model causes a somehow synchronzed materal flow,
Smulaton and optmzaton of supply chans 43 the measured delays are much lower now. Therefore, the cost of the soluton of the lnear model n the second teraton decreases. After three teratons the smulaton and the lnear model have converged to the same soluton. 5.1 Determnstc problems wth fxed costs In order to verfy the qualty of the solutons, we create a set of 12 examples. For these test nstances we consder a smple supply chan consstng of three actors (a suppler, a producer and a customer) and a tme horzon of 30 perods. For transportaton of products two transport modules are used, whch connect the suppler and the producer as well as the producer and the customer. Two types of products are demanded by the customer: product 1 whch s provded by the suppler and sent va the producer to the customer and product 2 whch s manufactured by the producer usng product 1 as a raw materal. The cost structure s as follows: 2 The transportaton costs consst of fxed costs per delvery, whch are subject to a step functon. A transport costs 100, 200 or 300 monetary unts, dependng on the amount delvered. The costs of producton and transfer are separated nto varable costs and fxed costs. The varable producton costs are set to 30 and the fxed part s 50 monetary unts. Transferrng products costs 15 per product unt plus a fxed part of 10. Delayed delveres are penalzed by 100 monetary unts per product unt and perod. Concernng the demand at the customer we dstngush between nstances wth hgh demand and others wth low demand. The dfference les n the frequency of orders sent off by the customer. In hgh demand cases the occurrng orders n each perod are around the maxmum possble quanttes whch could be delvered consderng the capactes of the suppler and the producer. In low demand models the ordered amounts cover approxmately 70% of the possble delveres n each perod. Instances D1-L to D5-L (see Table 2) represent fve dfferent realzatons for low demand models. Accordngly, D6-H to D10-H correspond to fve dfferent realzatons of hgh demand models. The last two nstances, D1a-L and D6a-H, are modfcatons of nstances D1-L and D6-H, respectvely. The former ones consder exactly the same orderng amounts as D1-L and D6-H but the fxed costs for producton and transfer at the ntermedate node are ncreased to 1000 and 500, respectvely. For examples of ths sze t s possble to formulate an exact mxed-nteger model and determne the optmal soluton. The correspondng MIP formulaton conssts of 1,342 constrants, 1,080 contnuous and 300 bnary decson varables. For the smulaton approach the nonlnear parts are only consdered n the smulaton model tself, the connected lnear (non-nteger) model does not nclude any of them. See Table 2 for the resultng total costs of the smulaton and the optmal soluton (MIP). The gap between our SmLP approach the optmal solutons ganed by solvng the exact MIP formulaton vares between 0.44 and 3.46% and averages n 1.87%. As we would expect, for the two test nstances wth hgh fxed cost the gap ncreases. 2 All datasets are avalable at http://www.unve.ac.at/bwl/prod/download/scm-data.
44 Table 2 Comparson of total costs between our smulaton-based optmzaton approach SmLP and the exact MIP-model for determnstc test cases classfed by the occurrence of customer demand (H hgh demand, L low demand) MIP solutons marked wth (*) are best solutons found after one hour calculaton tme C. Almeder et al. Instance SmLP Exact MIP Dfference (%) D1-L 53640 52947 1.31 D2-L 55032 53860 2.18 D3-L 52626 52394 0.44 D4-L 54442 53600 1.57 D5-L 55198 54057 2.11 D6-H 59885 58830 1.79 D7-H 61257 60129 1.88 D8-H 59028 58347 1.17 D9-H 60403 59501 1.52 D10-H 61436 60365 1.77 D1a-L 63720 61587 3.46 D6a-H 76165 73761* 3.26 Average 77165 73760 1.87 For test nstances wth low demand the varaton of the gap seems hgher than for the test nstances wth hgh demand. But on average there seems no sgnfcant dfference between the results of those two groups of nstances. Based on the above results we may conclude that the error caused by neglectng fxed cost s low as long as the fxed costs are low compared wth other costs. If the fxed costs ncrease (relatve to the other costs), the nonlnear propertes should be consdered n the optmzaton model used for the SmOpt approach,.e. a SmMIP should be used nstead of the SmLP (see also the followng subsecton). 5.2 Test nstances wth stochastc transport delays In order to measure the qualty of our solutons n a stochastc envronment we prepare a set of test examples ncludng stochastc transportaton tmes. We compare our Sm- LP approach usng a smplfed lnear model wthout bnary varables wth an exact MIP-model. Ths MIP-model does not cover stochastc features and we have to provde estmated values of the transportaton tmes. Wthn the smulaton we consder unformly dstrbuted transportaton delays between 1 and 9 for transportatons from the suppler to the producer and between 1 and 5 for transportatons from the producer to the customer. For estmatng the delay parameters we perform runs usng 90%-, 70%-, and 50%-quantles. The correspondng transportaton delays for the MIP-model are set accordng to the used quantle. For the small test cases there would probably be no notceable dfference between the results of a 99%- and a 90%- quantle. Therefore, we test a hgh quantle (90%, rsk averse), the average value (50%), and some ntermedate value (70%). The maxmum runtme s set to 30 mnutes for the MIPmodel,.e. we report the best soluton found after ths tme lmt, whle the smulaton approach converges after a few seconds. For the smulaton approach we agan process 8 teratons, each consstng of 5 smulaton runs and one LP computaton. Fnally the soluton of the MIP-model and the soluton of the SmLP are evaluated by performng
Smulaton and optmzaton of supply chans 45 Table 3 Dfference of the mean total costs of 20 runs between the SmLP and SmMIP method and the soluton found by a determnstc MIP model classfed by the occurrence of customer demand (H hgh demand, L low demand) Instance MIP SmLP SmMIP Cost Quant. (%) Cost Quant. (%) Dff. (%) Cost Quant. (%) Dff. (%) S1-L 66400 90 62601 90 5.72 61637 90 7.17 S2-L 61338 90 60635 90 1.15 60282 90 1.72 S3-L 63323 90 63566 70 0.38 63618 70 0.47 S4-L 63122 90 64067 90 1.50 64060 90 1.49 S5-L 60954 90 62399 90 2.37 62229 90 2.09 S6-H 72485 90 72342 90 0.20 70871 90 2.23 S7-H 70928 90 70751 90 0.25 71040 90 0.16 S8-H 73257 90 77537 70 5.84 74999 70 2.38 S9-H 73501 90 74637 70 1.55 72845 90 0.89 S10-H 71606 90 70230 90 1.92 73934 90 3.25 S1a-L 71511 90 71686 90 0.25 70350 90 1.62 S6a-H 88442 70 90582 90 2.42 88442 70 0.00 Av.-L 64441 64159 0.39 63696 1.08 Av.-H 75037 76013 1.24 75355 0.44 Average 69739 70086 0.42 69526 0.32 The total costs are reported n the columns Cost. The quantle whch lead to the best results s reported n the column Quant. The dfference wth respect to the soluton of the determnstc MIP s denoted n column Dff. 20 ndependent smulaton runs. Furthermore, we replace the smplfed lnear model wth the MIP model (SmMIP approach) and perform the same tests. The results for all three methods are dsplayed n Table 3, where negatve percentage values mply that the smulaton acheves a better result than the exact MIP-model. The results of the SmLP approach, where we combne the pure LP model wth the smulaton model, are on average slghtly worse compared wth the determnstc MIP approach. For the low demand cases alone we can observe a small mprovement. Consderng the varaton of the results these dfferences are not sgnfcant. Furthermore, we test a second approach, the SmMIP approach, where all nonlnear features of the model are also consdered n the optmzaton part whch s represented by a MIP model. So the dfference between the SmMIP and the exact approach s only, that n the frst case the parameters are estmated based on smulaton experments and n the latter case the parameters are determned usng the known dstrbuton functons. Here we see that the soluton qualty can be rased for 9 out of 12 test nstances and on average the result s slghtly better than the determnstc MIP approach. Even f for some nstances the 70%-quantle yelds the best results, the 90%- quantle leads to only slghtly hgher costs. Usng the 50%-quantle,.e. the expected value, always caused much hgher costs. Addtonally we analyze the varaton of the 20 fnal smulaton runs for each method. There s no sgnfcant dfference of the varaton for all methods. The coeffcent of varaton for the total costs s always around 0.07.
46 C. Almeder et al. For larger test nstances t would not be possble to solve the MIP or to apply the Sm- MIP method. In a prelmnary study (cf. Mtrovc 2006) we focused on that ssue and tred to fnd the approxmate lmts of solvng MIP formulatons of supply chan problems by the means of three state-of-the-art LP/MIP-solvers. The tests were conducted on a PC (Intel P4 2.4 GHz, 1 GB RAM) usng Wndows 2000. We used a set of dfferent szed supply chan network problems consderng fxed costs n transportaton, producton and transfer. The best performng solver succeeded n solvng problems wth 20 supply chan actors, 8 products, 5 perods and 2 transportaton modes, consderng 3,360 bnary varables before and 250 bnary varables after presolvng. The next n sze, whch ncluded 10 products nstead of 8, could not be solved wthn a tme lmt of one hour. Thus, there s a trade-off between a good approxmaton resultng from MIP-models or fast computatonal tmes. Defntely, mportant decsons nvolvng hgh fxed costs should be consdered wthn the optmzaton model of our SmOpt method. 5.3 Quantle tests on larger nstances In addton to the small nstances used n the prevous subsectons we generate a set of 12 nstances representng larger supply chan networks. Usng these test nstances we analyze the nfluence of the quantle on the results f the uncertanty s concentrated n a specfc part of the supply chan. The sze and structure of these test cases s shown n Fg. 5. Ths fcttous supply chan network conssts of 10 actors: 3 supplers, 4 producton nodes, and 3 customers. The ntermedate nodes are separated nto two layers and all of them are authorzed to produce and also transfer products. All actors are connected sup 1 cust 1 prod 1 prod 3 sup 2 cust 2 prod 2 prod 4 sup 3 cust 3 Fg. 5 Exemplary supply chan network. For smplcty the transport modules have been omtted
Smulaton and optmzaton of supply chans 47 Table 4 Total costs of SmLP for test example wth ten supply chan actors and stochastc transportaton delays at the begnnng of the supply chan The results for the 90%-quantle are taken as basc values. For the remanng quantles the dfference to the correspondng basc value s gven. The S ndcates that there s more stochastcty close to the suppler Instances Quantle 90% 70% 50% L1-L-S 274239 0.19% 11.20% L2-L-S 274241 9.98% 6.00% L3-L-S 274995 5.76% 10.85% L4-L-S 275366 5.83% 11.51% L5-L-S 273214 1.33% 9.80% L6-H-S 270286 2.72% 10.10% L7-H-S 270491 2.23% 4.51% L8-H-S 270438 0.59% 10.92% L9-H-S 267766 2.71% 8.65% L10-H-S 270155 0.57% 10.12% L1a-L-S 333772 2.77% 4.04% L6a-H-S 346953 1.42% 4.52% Total Avg. 283493 2.78% 8.52% by one transportaton mode. The customers request 4 dfferent products. Products 1 and 2 are on the one hand fnal products, whch have to be delvered to the customers and on the other hand raw materals used to produce products 3 and 4. Frst we consder the case wth the stochastcty concentrated at the begnnng of the supply chan. Hence, for the connecton between the supplers and the frst layer of producton stes we assume stochastc transportaton tmes, whch are unformly dstrbuted between 1 and 5. The transportaton tmes between the two layers of producton nodes are unformly dstrbuted between 1 and 3. For the remanng lnks we assume determnstc transportaton tmes of 1. The costs functons for producton and transport consst of fxed costs and varable costs. If all bnary decsons would be consdered n a MIP-model, ths would lead to more than 3,800 bnary varables, whch s beyond the sze of problems we could solve wth the best MIP/LP-solvers wthn several hours. In comparson, our SmLP algorthm takes about 12 mn for one test nstance to converge to a soluton. We evaluate three dfferent values for the quantle used for the estmaton of the delay parameters: 90, 70, and 50%. See Table 4 for the results. In ths case t seems that the 90%-quantle s most robust choce, although the 70%- quantle delvers only slghtly worse results on average and n some cases even better ones, whereas the 50%-quantle leads to the worst results for all nstances. Due to the fact that there s less stochastc near to the customer, t s possble to reduce the safety factors for the delays to some extent wthout ncreasng the costs too much, because lost tme at the begnnng of the supply chan can be made up at the end. We also conduct experments where the transportaton delays at the end of the supply chan are stochastc,.e. the connecton between producton nodes and customers are unformly dstrbuted between 1 and 5. Transportaton tmes between the two layers of producton nodes are agan unformly dstrbuted between 1 and 3. Remanng transportaton tmes are set to 1. The correspondng results are summarzed n Table 5.
48 Table 5 Total costs of SmLP for test example wth ten supply chan actors and stochastcty concentrated near the end of the supply chan The results for the 90%-quantle are taken as basc values. For the remanng quantles the percentage dfference to the correspondng basc value s gven. The C ndcates that there s more stochastc at the customer C. Almeder et al. Instance Quantle 90% 70% 50% L1-L-C 263957 11.70% 43.96% L2-L-C 263252 16.81% 45.02% L3-L-C 263948 13.79% 48.81% L4-L-C 263114 12.65% 50.60% L5-L-C 263707 13.98% 51.41% L6-H-C 258440 18.11% 52.09% L7-H-C 257963 14.60% 59.29% L8-H-C 260606 12.67% 41.68% L9-H-C 258762 24.74% 52.44% L10-H-C 258936 17.52% 43.16% L1a-L-C 335837 7.15% 32.56% L6a-H-C 335973 9.61% 32.95% Total Avg. 273708 14.45% 46.16% For these nstances the best choce would be to use the hghest safety factor (90%), because f there are delays at the end of the supply chan, there s no chance to catch up. For the test nstances L1a-H-C and L6a-H-C wth hgh fxed cost, we also apply a SmMIP method where we nclude only the hgh fxed-charge producton costs. The low fxed-charge transportaton costs are stll neglected. For all quantles the Sm- MIP method delvers slghtly lower costs (L1a-L-C: 4.48%/ 10.86%/ 1.19% for 90%/70%/50%-quantles; L6a-H-C: 3.47%/ 2.96%/ 0.37% for 90%/70%/50%- quantles) but the calculaton tmes are more than fve tmes longer. If we assume stochastc transportaton tmes for the whole supply chan (all transportaton delays are unformly dstrbuted between 1 and 5), the results are smlar to those n Table 5,.e. the 90%-quantle s always the best choce. 6 Conclusons In ths paper we have presented a new approach that combnes the advantages of complex smulaton models and abstract optmzaton models. We have shown that our method s able to generate compettve solutons quckly, even compared wth tradtonal plannng approaches that are much more tme consumng. Our nvestgatons can be summarzed as follows: In many cases the SmLP method seems to be a good trade-off between soluton qualty and computatonal tme. If the nonlnear elements n the model are domnatng t s better to apply the SmMIP approach and consder these nonlneartes n the optmzaton model as along as the computatonal tme for solvng the optmzaton model s acceptable.
Smulaton and optmzaton of supply chans 49 Furthermore, we nvestgated the mpact of safety tmes for delays on the soluton qualty. If we use the 90%-quantle, we can generate robust plans, but for specfc stuatons we mght get better results wth less safety tme. Only for the case f stochastc s near the customer, then the 90%-quantle s clearly the best. Nevertheless, the choce of the quantle depends on the structure of the supply chan and has to be fne-tuned n each case. Usng the 50%-quantle,.e. the expected values for the delays, always leads to pure results. If the uncertanty s concentrated far away from the customer, the cost ncrease by usng the expectaton value s about 10% whereas the ncrease s almost 50% f the uncertanty occurs close to the customer. Further research for dfferent aspects of ths method s stll possble and necessary. The aggregaton step and the generaton of new decson rules s an open feld. One possblty s to nterpret the soluton of the optmzaton model only as a target strategy and use adaptve decson rules to approxmate ths target strategy n an uncertan envronment. The use of senstvty results of the optmzaton model mght lead to mproved decson rules. Further nvestgatons are possble for the boundares between the smulaton and the optmzaton model. The queston, whch aspects should be ncluded n the optmzaton model, s not completely answered yet. If more complex models are used, other fast soluton methods (e.g., heurstcs, metaheurstcs, etc.) should be taken nto consderaton. We conclude by answerng the queston posed n the ttle of ths paper: smulaton and optmzaton are complementary approaches and t s worthwhle combnng them. Acknowledgment We wsh to thank Martn Grunow and two anonymous referees for ther valuable comments on ths manuscrpt. Appendx A: MIP formulaton for fxed-charge transportaton cost The objectve (20) of the optmzaton model can be transformed nto a mxed-nteger program consderng ( for example ) fxed transportaton costs. In case, the transportaton cost functons v C p v j x p j (t) can be wrtten as follows: ( ) { v v C p v j x p j (t) c p = j f v x p j (t) >0 0 otherwse, p, t,v. (22) In order to capture ths stuaton, t s necessary to ntroduce bnary decson varables v Ɣ p j (t) whch ndcate f there s postve transportaton. So by addng the followng constrants: v x p j (t) G vɣ p j (t), p, t,v (23) t s possble to formulate the transportaton costs as the lnear functons
50 C. Almeder et al. v C p j (t) = v c p j v Ɣ p j (t), p, t,v. (24) The resultng mxed-nteger lnear program ncludes now J I P T V bnary decson varables. A smlar approach can be used for modelng step functons lke ( ) v C p v j x p j (t) = v c p v j d p j f v x p j (t) (v X p j, v Y p f v x p j (t) (0, v X p j ] 0 otherwse j ], p, t,v. (25) Here we need 2 dfferent bnary decson varables v Ɣ p j (t) and v p j (t) to represent ths stuaton. If we add two addtonal constrants v x p j v x p (t) G vɣ p j (t), p, t,v, (26) j (t) G v p j (t) + v X p j (t), p, t,v, (27) the cost functons can be wrtten as ( v C p j (t) = v d p j v Ɣ p j (t) + v c p j v dj) p v p j (t), p, t,v. (28) The resultng mxed-nteger lnear program ncludes now 2 J I P T V bnary decson varables. Appendx B: Notaton Notaton used for the optmzaton model J set of locatons J = J S J I J C j J S raw-materal suppler (startng nodes) j J C customer (end nodes) j J I nodes between suppler and customer P set of products V set of transportaton modes T number of perods Decson varables m p (t) amount of product p (product p s the end product of the producton process at locaton ) that starts to be produced at locaton n perod t u p (t) amount of product p that starts to be transacted n locaton n tme perod t v x p j (t) flow of product p from locaton to locaton j wth transportaton mode v (sent away n perod t)
Smulaton and optmzaton of supply chans 51 Costs, delays, and general parameters a p factor ndcatng the amount of capacty unts requred to produce one unt of product p at locaton α p (p ) amount of product p requred to produce one unt of product p at locaton n b p (t) amount of backorders of product p at customer n perod t v C p j ( ) transportaton cost functon of product p transported from locaton to locaton j wth transportaton mode v v C j (t) maxmum transportaton capacty of transportaton mode v on the way from locaton to locaton j prod C (t) maxmum producton capacty at locaton n perod t ta C (t) maxmum transacton capacty at locaton n perod t nvn Cap p (t) maxmum amount of product p that can be held n the nbound nventory of ntermedate n perod t nvout Cap p (t) maxmum amount of product p that can be held n the outbound nventory of ntermedate n perod t v Cap p j (t) amount of product p that transportaton mode v can transport from locaton to locaton j n perod t prod Cap p (t) amount of product p that can be produced at locaton n perod t ta Cap p (t) amount of product p that can be transacted at locaton n perod t v c p j cost factor used n case of lnear transportaton costs for delveres of product p between locaton and locaton j wth transportaton mode v D p d p (t) δ p n f p j (t) out f p v g p n H p out H p n h p out h p j (t) ( ) ( ) n L (t) out L (t) n l p (t) out l p (t) q p R p ( ) demand for product p at locaton n perod t factor ndcatng the amount of capacty unts requred to transact one unt of product p at locaton amount of perods requred to produce product p at locaton amount of product p arrvng at locaton j n perod t amount of product p sent away at locaton j n perod t factor ndcatng the amount of capacty unts requred to transport one unt of product p wth transportaton mode v nbound nventory cost functon for product p at locaton outbound nventory cost functon for product p at locaton cost factor used n case of lnear nventory costs for the nbound nventory of actor and for product p cost factor used n case of lnear nventory costs for the outbound nventory of actor and for product p maxmum capacty of nbound nventory at locaton n perod t maxmum capacty of outbound nventory at locaton n perod t nbound nventory level of product p at locaton n perod t outbound nventory level of product p at locaton n perod t factor ndcatng the amount of capacty unts requred to hold one unt of product p at the nventory of locaton penalty cost functon at locaton for product p
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